2009-02-18
Today we'll be tuning a bit of music attributed to Orlande de Lassus.
Here's the ASCII score
Meantone
G----G A Bb---Bb A----A G.....A----A G----G F# E F# G
D E F----F----G----G F----F E D E D----D D....... D
B C----C D----D C----C----C-----C B----B A....... B
G C F Bb---Bb C----C----C-----C D----D D....... G
Dashed lines indicate ties; dotted lines indicate held notes.
You can compare the optimized meantone to 12-ET.
To put this in 5-limit JI, we will have to introduce compromises into
the melodic structure. There are several ways to do this:
Drift
1----1 5/9 /27--/27 5/9--5/9 1 5/9---5/9 5/81--. 5/81
3 5/3 /9---/9 1----1 /9---/9 5/3 5/27--. . 5/27
5 /3---/3 5/27~~3 /3---/3---/3---/3 25/81--. 5/9 25/81
1 /3 /9 /27--/27 /3---/3---/3---/3 5/27--. . 5/81
*
Here we let the piece end on a tonic one syntonic comma below where it
started. I haven't rendered it but, trust me, it isn't pleasant.
The asterisk indicates that we're also taking a harmonic purity hit,
with 32/27 standing in for 6/5.
The tildas (~~) indicate a tie that doesn't end on the same pitch it
started. Single dots are shorthand for the previous pitch on a line.
Lumma2005
1----1 5/9 /27--/27 5/9--5/9 1 5/9~~~~9 1----1 1
3 5/3 /9---/9 1----1 /9---/9 5/3 3----3 3 3
5 /3---/3 5/27~~3 /3---/3---/3---/3 5----5 9 5
1 /3 /9 /27--/27 /3---/3---/3---/3 3----3 3 1
*
Same as above, except we take another "comma shift" (the tildas) to get
the piece to end on 1/1.
deVelde2009a
1----1 5/9 3/5--3/5 5/9--5/9 1 9-----9 1----1 1
3 5/3 /9---/9 1----1 /9---/9 27 3----3 3 3
5 /3---/3 3----3 /3---/3---/3---/3 5----5 9 5
1 /3 /9 3/5--3/5 /3---/3---/3---/3 3----3 3 1
* *
This one gets rid of the shifts at the cost of a more severe consonance
hit near the beginning (40:27) and an additional one at the end.
To my ear, none of these JI versions are acceptable. One would never
hear a choir perform this way.
The meantone version at the top sounds passable, but can we do better?
Well, yes. Nicola Vicentino proposed a way in 1555. His solution is
shown near the bottom of Joe Monzo's page on the topic.
I'm mirroring Joe's MIDI file here. To my ears, we've finally got
something that sounds right, just as you'd sing it.
~